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For what value of p can the function f(x) = { ((4^x - 1)^3)/(sin x · [log|1 + x^2/3|]^p), x ≠ 0; 12(\log 4)^3, x = 0 } be made continuous at x = 0?

1. 1
2. 2
3. 3
4. None of these

Correct answer: 2

Solution

To ensure continuity at x = 0, the limit of f(x) as x approaches 0 must equal f(0). Evaluating the limit leads to a form that requires balancing the growth of the numerator and the denominator, which is achieved when p = 2, making the function continuous at that point.

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